The triple difference causal inference framework is an extension of the well-known difference-in-differences framework. It relaxes the parallel trends assumption of the difference-in-differences framework through leveraging data from an auxiliary domain. Despite being commonly applied in empirical research, the triple difference framework has received relatively limited attention in the statistics literature. Specifically, investigating the intricacies of identification and the design of robust and efficient estimators for this framework has remained largely unexplored. This work aims to address these gaps in the literature. From the identification standpoint, we present outcome regression and weighting methods to identify the average treatment effect on the treated in both panel data and repeated cross-section settings. For the latter, we relax the commonly made assumption of time-invariant covariates. From the estimation perspective, we consider semiparametric estimators for the triple difference framework in both panel data and repeated cross-sections settings. These estimators are based upon the cross-fitting technique, and flexible machine learning tools can be used to estimate the nuisance components. We demonstrate that our proposed estimators are doubly robust, and we characterize the conditions under which they are consistent and asymptotically normal.

翻译：三重差分因果推断框架是广为人知的双重差分框架的扩展。该框架通过利用辅助领域的数据，放松了双重差分框架中的平行趋势假设。尽管三重差分框架在实证研究中被广泛应用，但统计学文献对其关注相对有限。具体而言，关于该框架识别机制的深入探讨以及稳健高效估计量的设计，在很大程度上仍未得到充分研究。本文旨在填补文献中的这些空白。从识别角度出发，我们提出了结果回归和加权方法，以识别面板数据和重复横截面数据设置下处理组的平均处理效应。对于后者，我们放宽了通常假设的时不变协变量条件。从估计角度出发，我们为面板数据和重复横截面数据设置下的三重差分框架考虑了半参数估计量。这些估计量基于交叉拟合技术，并可使用灵活的机器学习工具来估计干扰项。我们证明了所提出的估计量具有双重稳健性，并刻画了其具有一致性和渐近正态性的条件。